# On the Melting Thresholds of Semiconductors under Nanosecond Pulse Laser Irradiation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{m}[5,6,26,27,28].

## 2. Model Description

^{−12}s for silicon and germanium [7], and similar time scales for the other materials under study. In addition, as we consider relatively low surface temperatures, below and near the melting point, we disregard evaporation phenomena, which, however, may slightly affect the melting process for compound semiconductors [30,31]. For correct simulations of laser–matter interaction processes, material thermophysical and optical properties (and their temperature dependences) are of fundamental importance. The material parameters used in the presented calculations are summarized in Appendix A, Table A1, Table A2, Table A3, Table A4, Table A5, Table A6, Table A7, Table A8, Table A9, Table A10, Table A11, Table A12, Table A13, Table A14, Table A15, Table A16, Table A17, Table A18 and Table A19.

_{p}, ρ, L

_{m}, T

_{m}and κ are, respectively, the heat capacity, the density, the latent heat of fusion, the melting temperature and the thermal conductivity of the sample material. Energy supplied by the laser is represented by the source term $S\left(z,t\right)$ as:

_{0}and τ

_{L}being the peak fluence and the pulse duration.

^{*}, are unknown at the time moment t

_{f}, and T (without asterisk) corresponds to the known temperature at the time moment t

_{f−1}= t

_{f}− Δt.

_{m}, and its ratio to the enthalpy of melting can be interpreted as a molten fraction in a computational element. In the presented calculations, we apply the method of through calculation without explicit selection of the phase interface [32,33]. According to this method, the melting process is smoothed over a symmetric interval of a width of a few Kelvins around the melting point. Melting starts at a slightly lower temperature than T

_{m}, reaches the melting point at the fraction of molten material of 0.5, and ends at a slightly higher temperature than T

_{m}. In the interval of melting, a δ function is added to the heat capacity term to account for absorption/release of the fusion heat at the melting/solidification front:

_{s}, γ

_{l}represent a property of material in solid and liquid state, respectively, and η is the fraction of molten material. For the sake of simplicity, the change in density upon melting is not taken into account so that the value of the solid-state density is also kept for the liquid state.

_{th}. According to the δ-function approach (Equations (1) and (6)), this occurs at ~1–2 K bellow the tabulated melting point and thus the edge of beginning of melting is blurred. However, from analyzing our simulation data, it follows that the position of F

_{th}within the interval of melting has a minor effect on its resulting value. Furthermore, taking into account the ambiguity of F

_{th}reported in the literature, this aspect plays only a small role.

## 3. Results and Discussion

_{th}values can be determined differently from the method used in this study. For example, in Ref. [39], the calculated melting threshold for CdTe was set 8% higher than the laser fluence needed for reaching T

_{m}. Time resolved reflectometry (TRR) measurements performed by the authors did not show an increase in reflectivity at the intensity corresponding to reaching the melting point in the calculations. Thus, as the melting threshold, the authors consider the intensity at which the sample surface layer is molten to the depth of laser radiation absorption. Experimentally measured values of the melting threshold fluence typically include a transition interval where localized melting occurs, giving rise to an increase in the reflectivity above the values of solid-state surface reflectivity [13,29]. In numerical simulations, the determination of the damage threshold strongly depends on the used material properties [40]. Their choice can be considered as the main source of inaccuracy of the modeling results because we use a very accurately verified implicit algorithm for numerical solution (Section 2). Below we have surveyed the literature for the optical and thermophysical parameters of the studied semiconductors. The most relevant parameters are given in Appendix A.

#### 3.1. Silicon

_{p}, we took the data from Ref. [41] with a stronger variation over the range of solid-state temperatures than the dependence used for c-Si in Ref. [5]. The c-Si thermal conductivity was approximated by the expression from the measured data reported in [42]. For the liquid state, we use c

_{p}= 910 J/(kg·K) and κ = 50.8 + 0.029(T − T

_{m}) W/(m·K) [5]. The reflectivity and the absorption coefficient for c-Si are temperature dependent and given by the relations presented in [43]. The optical properties of molten silicon are described according to the calculated data for 694 nm [44] and measured data for 352 nm [45]. The data on the properties for solid and liquid silicon used in the present modeling are summarized in Table A1, Table A2, Table A3 and Table A4 of Appendix A.

_{th}values presented in [6]. The experimental investigations reported in [37] for 532 nm ns laser irradiation give an interval of increasing reflectivity between 330 and 380 mJ/cm

^{2}. The value of 380 mJ/cm

^{2}was identified as a threshold for reaching a high reflectivity (~70%), probably indicating melting to a depth of approximately one optical skin layer (~10 nm), and the maximum reflectivity of 73% was observed at 450 mJ/cm

^{2}. A similar value of around 400 mJ/cm

^{2}was determined as a threshold for melting based on time-resolved reflectivity measurements [38]. The theoretical calculations [6] give higher values for the melting thresholds than those calculated in this work and measured in [37,38]. The main reason for this can be seen in the difference in the absorption coefficient change with temperature. For instance, the absorption coefficient of c-Si at a 694 nm wavelength at temperatures close to the melting point is approximately five times larger in our case (taken from [43]) than in calculations presented in [6]. Note that the analytical theory [27] predicts an even higher threshold fluence for Si under the same irradiation conditions (694 nm, 30 ns, see Table 1). As a whole, our modeling data for Si are in reasonable agreement with most of the published data, thus demonstrating that our model approach can be used for other semiconducting materials. The calculations with various pulse durations, τ

_{L}, show that the melting threshold increases with τ

_{L}proportionally to app τ

_{L}

^{0.2}(Table 1), i.e., the dependence is considerably weaker than the ∝ τ

_{L}

^{0.5}dependence predicted for the evaporation threshold for fairly long (nanosecond and longer) laser pulses [46].

#### 3.2. Germanium

^{2}as a value, at which the rise of reflectivity was detected corresponding to the observable melting. Numerical simulations using the finite difference method were also carried out in Ref. [10], and the obtained melting threshold was claimed to be “practically identical” to the measured one (although the method for threshold determination in the simulation was not specified). The authors used experimentally measured values of reflectivity and absorption from Refs. [9,47], which are in good agreement with the optical constants we derived from measurements reported in [48] and also confirmed in [49]. Our calculations give F

_{th}= 370 mJ/cm

^{2}(Figure 1a, Table 1), which is in good agreement with the data [10], particularly taking into account that the increase in reflectivity detected in [10] assumes a significant fraction of molten germanium and thus a slightly higher fluence than that needed to reach the melting temperature at the surface. Near the melting threshold, the calculated melt fraction reaches a maximum after a delay of approximately 20 ns relative to the moment of laser peak intensity (Figure 1a), which is also in agreement with the measurements [10]. With the known properties of liquid Ge (Table A7 and Table A8), we have performed simulations for F > F

_{th}, which are again in good agreement with the measured durations of a high reflectivity stage corresponding to molten germanium [10] (Figure 1b). It should be mentioned that different values are reported for the thermal conductivity of liquid Ge. In the modeling, we use the value of 29.7 W/(m·K) [50], while in Ref. [51], κ = 43 W/(m·K) was measured. The calculations performed for various pulse durations demonstrate a stronger τ

_{L}dependence than that for silicon, close to the ∝ τ

_{L}

^{0.5}dependence (Table 1).

#### 3.3. Gallium Arsenide

_{p}, the data from [52] were used, which are also in a good agreement with the data reported in [53]. Optical properties were taken from measurements [54], which are also in a good agreement with [55]. The absorption and reflection coefficients were calculated from the refractive index and the extinction coefficient and taken as temperature-independent. The material properties used in the simulations are presented in Table A9, Table A10, Table A11 and Table A12 of Appendix A.

^{2}, which corresponded to the situation when a ~65-nm-thick surface layer was molten [12]. In our simulations, this fluence of 300 mJ/cm

^{2}results in a melting depth of 13 nm, while the melting threshold corresponding to reaching the melting point on the sample surface is 265 mJ/cm

^{2}(see Figure 2 for comparison). The difference in the melting depth can be attributed to two factors. First, the authors [12] extrapolated the temperature-dependent absorption coefficient for the solid-state GaAs from the room temperature util the melting point, which appears to be questionable. In our simulations, we use constant but reliable data on the optical absorption and reflectivity of molten GaAs at the wavelength of the ruby laser [48]. The reflectivity coefficient of liquid GaAs in both Ref. [12] and this work was adopted from [13], R = 0.67. The second factor may be related to using an explicit numerical scheme in Ref. [12], whose approximation to the initial equations often represent a challenge.

#### 3.4. Cadmium Telluride

^{2}as the melting threshold. In their simulations, this value corresponds to the molten layer with a thickness of the laser absorption depth. Their TRR measurements detected an abrupt although small rise of the reflectivity at a laser fluence of 48–50 mJ/cm

^{2}. These results are in excellent agreement with our simulations (Table 1). Indeed, for F = 50 mJ/cm

^{2}, our model gives the depth of the molten layer of 7 nm, very close to the absorption depth of CdTe at 248 nm (~9 nm). According to our definition of the melting threshold, achieving the melting temperature at the very surface of the irradiated sample, the calculated threshold is slightly lower, 46 mJ/cm

^{2}(Table 1).

#### 3.5. Indium Phosphide

^{2}was identified as the damage threshold. In our simulations, we have obtained a threshold value of 106 mJ/cm

^{2}, which can be considered as a good agreement taking into account that there are no fitting parameters in our model. It should be mentioned that, although laser processing of InP is a common technique in its industrial applications, the thermophysical parameters at enhanced temperatures are still not well studied. Thus, several sets of data are available for the heat capacity of solid InP; see, e.g., [61]. The major problem is that measurements of the thermophysical properties at enhanced temperatures are affected by the high vapor pressure of phosphorous due to its high volatility. The thermal conductivity and the specific heat of molten InP are given in [53]. The reflectivity and absorption are calculated from data provided in [48]. Optical properties are taken as temperature independent and are considered the same for both solid and liquid state. In reference article [30], ablation of compound semiconductors is studied, and a model that takes into account evaporation of their components gives the melting threshold. Our result, which disregards this effect, gives F

_{th}that is approximately 10% higher.

#### 3.6. Generalization of the Damage Threshold Data into a Predictive Dependence

^{2}for CdTe to almost 1 J/cm

^{2}for Si (Table 1). The irradiation conditions also affect the threshold values, which are generally smaller for shorter laser wavelengths and pulse durations. It is very attractive to generalize the obtained results in terms of a unified parameter combining the basic material properties (thermophysical and optical) in order to be able to predict the ns-laser-induced melting thresholds, at least approximately, without performing detailed simulations.

_{p}is the thermal diffusivity and ΔH = L

_{m}+ c

_{p}(T

_{m}− 300) is the total energy needed to heat the sample to the complete melting state from room temperature, A similar parameter was introduced in [46] as an evaporation threshold under ns-laser ablation (assuming naturally by ΔH in Equation (8) the specific heat for evaporation instead of that for melting and omitting the 2/(1 − R) factor).

_{B}parameter, Equation (8), evaluated for all the studied materials using their room-temperature properties. All the data are nicely grouped around a straight line in the logarithmic plot. This clear correlation is rather surprising for such a simplified generalization approach when the material absorption coefficient and temperature dependencies of thermophysical properties are not taken into account. The least square fitting line in Figure 4 is described by a power law, F

_{th}≈ 0.05 P

_{B}

^{1.16}, which can be used for a rough estimation of the melting threshold of semiconductors based on their basic room-temperature properties.

_{B}predicts a growth of the melting threshold with the laser pulse duration as τ

_{L}

^{0.5}. However, as was noticed above, this is not always the case according to our simulations. Some semiconductors (Ge, CdTe, InP) follow closely the τ

_{L}

^{0.5}dependence, while others (Si, GaAs) demonstrate weaker dependencies (Table 1 and Figure 4). This finding agrees with calculations based on the theory in [26], which also predicts a nearly τ

_{L}

^{0.5}dependence for CdTe over a wide range of pulse durations [28] and a much weaker dependence for Si [27]. This is probably mainly due to a difference in the thermal diffusivity, D, of the materials. Thus, at room temperature, D ≈ 0.8 cm

^{2}/s for Si, and it is around 0.35 cm

^{2}/s for Ge, InP and CdTe. A higher thermal diffusivity results in a higher heat diffusion length and smaller in-depth temperature gradients and thus in a lower heat flow from the surface at an increased pulse duration. The temperature dependencies of material parameters (included in our model simulations) can additionally affect the pulse duration dependence of the melting threshold.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Silicon**

Property | Value | Ref. |
---|---|---|

ρ, g/cm^{3} | 2.328 | |

T_{m}, K | 1688 | |

L_{m}, J/kg | 1.826 × 10^{6} | [62] |

c_{p}, J/kg K | 847.05 + 118.1 × 10^{−3} T − 155.6 × 10^{5} T^{−2} | [42] |

κ, W/mK | 97269 T^{−1.165} (300 < T < 1000)3.36 × 10 ^{−5} T^{2} − 9.59 × 10^{−2} T + 92.25 (1000 < T < T_{m}) | [42] |

Property | Value, 532 nm | Ref. | Value, 694 nm | Ref. |
---|---|---|---|---|

n | 4.152 | [48] | 3.79 | [48] |

k | 0.051787 | [48] | 0.013 | [48] |

R | 0.374 | [43] | 0.34 + 5 × 10^{−5} (T − 300) | [43] |

α, 1/m | 5.02 × 10^{5} exp (T/430) | [43] | 1.34 × 10^{5} exp (T/427) | [43] |

Property | Value, 532 nm | Ref. | Value, 694 nm | Ref. |
---|---|---|---|---|

n | 3.212 | [45] | 3.952 | [44] |

k | 4.936 | [45] | 5.417 | [44] |

R | 0.693 | Calculated | 0.707 | Calculated |

α, 1/m | 1.1659 × 10^{8} | Calculated | 9.804 × 10^{7} | Calculated |

**Germanium**

Property | Value | Ref. |
---|---|---|

ρ, g/cm^{3} | 5.327 | |

T_{m}, K | 1211.4 | |

L_{m}, J/kg | 5.1 × 10^{5} | [50] |

c_{p}, J/kg K | 1.17 × 10^{−1} T + 293 | [50] |

κ, W/mK | 18,000/T | [50] |

**Gallium Arsenide**

Property | Value | Ref. |
---|---|---|

ρ, g/cm^{3} | 5.32 | |

T_{m}, K | 1511 | |

L_{m}, J/kg | 7.11 × 10^{5} | |

c_{p}, J/kg K | 8.76 × 10^{−2} T + 308.16 | [8] |

κ, W/mK | 30,890 T^{−1.141} | [8] |

Property | Value, 308 nm | Ref. | Value, 532 nm | Ref. | Value, 694 nm | Ref. |
---|---|---|---|---|---|---|

n | 3.7 | [48] | 4.13 | [48] | 3.78 | [48] |

k | 1.9 | [48] | 0.336 | [48] | 0.15 | [48] |

R | 0.42 | Calculated | 0.37 | Calculated | 0.338 | Calculated |

α, 1/m | 7.7 × 10^{7} | Calculated | 8.04 × 10^{6} | Calculated | 2.687 × 10^{6} | Calculated |

Property | Value, 308 nm | Ref. | Value, 694 nm | Ref. |
---|---|---|---|---|

R | 0.46 | [8] | 0.67 | [13] |

α, 1/m | 0.83 × 10^{8} | [63] | 2.687 × 10^{6} | Taken the same as for solid-state |

**Cadmium Telluride**

Property | Value | Ref. |
---|---|---|

ρ, g/cm^{3} | 5.85 | |

T_{m}, K | 1365 | |

L_{m}, J/kg | 2.09 × 10^{5} | [59] |

c_{p}, J/kg K | 3.6 × 10^{−2} T + 205 | [59] |

κ, W/mK | 1507/T | [57] |

Property | Value, 248 nm | Ref. | Value, 694 nm | Ref. |
---|---|---|---|---|

n | 2.63 | [60] | 3.037 | [60] |

k | 2.13 | [60] | 0.286 | [60] |

R | 0.406 | Calculated | 0.258 | Calculated |

α, 1/m | 1.1 × 10^{8} | Calculated | 5.179 × 10^{6} | Calculated |

**Indium Phosphide**

Property | Value | Ref. |
---|---|---|

ρ, g/cm^{3} | 4.81 | |

T_{m}, K | 1335 | |

L_{m}, J/kg | 3.4 × 10^{5} | [64] |

c_{p}, J/kg K | 2.33 × 10^{−2} T + 347 | [61] |

κ, W/mK | 1.215 × 10^{5} T^{−}^{1.324} | [61] |

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**Figure 1.**(

**a**) Surface temperature (solid lines) and molten fraction (dot-dashed lines) of germanium obtained in the modelling for different laser fluences (694-nm, 70-ns pulse). The laser temporal profile is shown by the dashed line. (

**b**) Comparison of the duration of increased reflectivity (data from experiment [10]) and the melt duration obtained in our simulations.

**Figure 2.**Comparison of the results obtained by modelling in Ref. [12] and in this work for GaAs irradiated by 694-nm, 15-ns laser pulses. (

**a**) The depth of molten material as a function of time for several laser fluences (reprinted with permission from Ref. [12]). (

**b**) The results of the present modeling for laser fluences of 265 mJ/cm

^{2}(corresponding to the defined melting threshold) and 300 mJ/cm

^{2}. The temporal evolution of the melt depth at 300 mJ/cm

^{2}is also given to compare with Ref. [12]. The laser pulse is shown by the dashed line.

**Figure 3.**(

**a**) The number of Ga and As atoms emitted from the GaAs surface irradiated by 694-nm, 20-ns laser pulses as a function of laser fluence as derived from mass spectrometric measurements (adapted with permission from Ref. [13]). Transient uneven and developed manifestations of increased reflectivity indicating the appearance of the liquid phase are marked by the shaded region and the solid line, respectively. Our simulated melting threshold is shown by a vertical dashed line. (

**b**) The simulated dynamics of the surface temperature with the identified melting threshold of 282 mJ/cm

^{2}. The laser pulse profile is shown by the dashed line.

**Figure 4.**Calculated melting thresholds of the studied semiconductors for different laser wavelengths as a function of the Bäuerle parameter, P

_{B}, Equation (8). The numbers above the points correspond to the laser pulse duration in nanoseconds. The line represents a power-law least-square fit of the data.

**Table 1.**The simulation results for the damage threshold fluence, F

_{th}, of the studied semiconductors in comparison with theoretical and experimental data reported in the literature. The experimental data are marked by asterisks *.

Material | λ, nm | τ, ns | F_{th}, mJ/cm^{2}This Work | F_{th}, mJ/cm^{2}Literature Data |
---|---|---|---|---|

Si | 532 | 18 | 355 | 395 [6], 330 * [37] |

30 | 423 | 474 [6] 350 [38] | ||

694 | 15 | 672 | 725 [6] | |

30 | 752 | 805 [6], 980 [27] | ||

70 | 900 | |||

Ge | 694 | 15 | 191 | |

30 | 255 | 300 [27] | ||

70 | 370 | 400 * [10] | ||

GaAs | 308 | 30 | 213 | 200, 200 * [8] |

532 | 15 | 184 | ||

694 | 15 | 265 | 300 [12] | |

20 | 282 | 250 * [13], 360 * [23], 240 [27] | ||

30 | 316 | |||

70 | 415 | |||

CdTe | 248 | 20 | 46 | 50, 50 * [39] |

694 | 15 | 68 | 78 [28] | |

30 | 80 | 98 [28] | ||

70 | 103 | 130 [28] | ||

InP | 532 | 7 | 106 | 97 [30] |

694 | 15 | 165 | ||

30 | 211 | |||

70 | 296 |

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**MDPI and ACS Style**

Beránek, J.; Bulgakov, A.V.; Bulgakova, N.M. On the Melting Thresholds of Semiconductors under Nanosecond Pulse Laser Irradiation. *Appl. Sci.* **2023**, *13*, 3818.
https://doi.org/10.3390/app13063818

**AMA Style**

Beránek J, Bulgakov AV, Bulgakova NM. On the Melting Thresholds of Semiconductors under Nanosecond Pulse Laser Irradiation. *Applied Sciences*. 2023; 13(6):3818.
https://doi.org/10.3390/app13063818

**Chicago/Turabian Style**

Beránek, Jiří, Alexander V. Bulgakov, and Nadezhda M. Bulgakova. 2023. "On the Melting Thresholds of Semiconductors under Nanosecond Pulse Laser Irradiation" *Applied Sciences* 13, no. 6: 3818.
https://doi.org/10.3390/app13063818